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Properties

Label 2-363-11.9-c1-0-10
Degree 22
Conductor 363363
Sign 0.569+0.821i0.569 + 0.821i
Analytic cond. 2.898562.89856
Root an. cond. 1.702511.70251
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.690 + 2.12i)2-s + (−0.809 + 0.587i)3-s + (−2.42 − 1.76i)4-s + (0.618 + 1.90i)5-s + (−0.690 − 2.12i)6-s + (−3.61 − 2.62i)7-s + (1.80 − 1.31i)8-s + (0.309 − 0.951i)9-s − 4.47·10-s + 3·12-s + (8.09 − 5.87i)14-s + (−1.61 − 1.17i)15-s + (−0.309 − 0.951i)16-s + (−1.38 − 4.25i)17-s + (1.80 + 1.31i)18-s + (−3.61 + 2.62i)19-s + ⋯
L(s)  = 1  + (−0.488 + 1.50i)2-s + (−0.467 + 0.339i)3-s + (−1.21 − 0.881i)4-s + (0.276 + 0.850i)5-s + (−0.282 − 0.868i)6-s + (−1.36 − 0.993i)7-s + (0.639 − 0.464i)8-s + (0.103 − 0.317i)9-s − 1.41·10-s + 0.866·12-s + (2.16 − 1.57i)14-s + (−0.417 − 0.303i)15-s + (−0.0772 − 0.237i)16-s + (−0.335 − 1.03i)17-s + (0.426 + 0.309i)18-s + (−0.830 + 0.603i)19-s + ⋯

Functional equation

Λ(s)=(363s/2 ΓC(s) L(s)=((0.569+0.821i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(363s/2 ΓC(s+1/2) L(s)=((0.569+0.821i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 363363    =    3⋅1123 \cdot 11^{2}
Sign: 0.569+0.821i0.569 + 0.821i
Analytic conductor: 2.898562.89856
Root analytic conductor: 1.702511.70251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ363(130,⋅)\chi_{363} (130, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 363, ( :1/2), 0.569+0.821i)(2,\ 363,\ (\ :1/2),\ 0.569 + 0.821i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(0.809−0.587i)T 1 + (0.809 - 0.587i)T
11 1 1
good2 1+(0.690−2.12i)T+(−1.61−1.17i)T2 1 + (0.690 - 2.12i)T + (-1.61 - 1.17i)T^{2}
5 1+(−0.618−1.90i)T+(−4.04+2.93i)T2 1 + (-0.618 - 1.90i)T + (-4.04 + 2.93i)T^{2}
7 1+(3.61+2.62i)T+(2.16+6.65i)T2 1 + (3.61 + 2.62i)T + (2.16 + 6.65i)T^{2}
13 1+(−10.5−7.64i)T2 1 + (-10.5 - 7.64i)T^{2}
17 1+(1.38+4.25i)T+(−13.7+9.99i)T2 1 + (1.38 + 4.25i)T + (-13.7 + 9.99i)T^{2}
19 1+(3.61−2.62i)T+(5.87−18.0i)T2 1 + (3.61 - 2.62i)T + (5.87 - 18.0i)T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1+(−3.61−2.62i)T+(8.96+27.5i)T2 1 + (-3.61 - 2.62i)T + (8.96 + 27.5i)T^{2}
31 1+(−25.0−18.2i)T2 1 + (-25.0 - 18.2i)T^{2}
37 1+(1.61+1.17i)T+(11.4+35.1i)T2 1 + (1.61 + 1.17i)T + (11.4 + 35.1i)T^{2}
41 1+(3.61−2.62i)T+(12.6−38.9i)T2 1 + (3.61 - 2.62i)T + (12.6 - 38.9i)T^{2}
43 1+4.47T+43T2 1 + 4.47T + 43T^{2}
47 1+(6.47−4.70i)T+(14.5−44.6i)T2 1 + (6.47 - 4.70i)T + (14.5 - 44.6i)T^{2}
53 1+(−1.85+5.70i)T+(−42.8−31.1i)T2 1 + (-1.85 + 5.70i)T + (-42.8 - 31.1i)T^{2}
59 1+(18.2+56.1i)T2 1 + (18.2 + 56.1i)T^{2}
61 1+(2.76+8.50i)T+(−49.3+35.8i)T2 1 + (2.76 + 8.50i)T + (-49.3 + 35.8i)T^{2}
67 1+12T+67T2 1 + 12T + 67T^{2}
71 1+(2.47+7.60i)T+(−57.4+41.7i)T2 1 + (2.47 + 7.60i)T + (-57.4 + 41.7i)T^{2}
73 1+(7.23+5.25i)T+(22.5+69.4i)T2 1 + (7.23 + 5.25i)T + (22.5 + 69.4i)T^{2}
79 1+(4.14−12.7i)T+(−63.9−46.4i)T2 1 + (4.14 - 12.7i)T + (-63.9 - 46.4i)T^{2}
83 1+(−2.76−8.50i)T+(−67.1+48.7i)T2 1 + (-2.76 - 8.50i)T + (-67.1 + 48.7i)T^{2}
89 1+14T+89T2 1 + 14T + 89T^{2}
97 1+(−0.618+1.90i)T+(−78.4−57.0i)T2 1 + (-0.618 + 1.90i)T + (-78.4 - 57.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.89586377851748224568865396089, −10.05348452180264706994765430231, −9.562941994497872692304665891305, −8.316925767634156724823786351147, −7.08636131381566549613954010335, −6.65347699744298434320375230247, −5.95499319061068649806064755413, −4.57766661514690438307229868508, −3.16544726887110387131289330136, 0, 1.74997079322915784401467708903, 2.91730361629439287662523984329, 4.30780903858763957364065878385, 5.76349084039087260304377611668, 6.60539953686446730429116473884, 8.488713369923183173420331367884, 8.944652116474531204865782281404, 9.925112454718506083738898296916, 10.55474462393294047640874709830

Graph of the ZZ-function along the critical line